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Problem 4.13PP
Consider the second-order system 
d s ) ? + 2 f7 + T *
We would like to add a transfer function of the fonn Dc(s) = in series with G(s) in s
unity feedback structure.
(a) Ignoring stability for the moment, what are the constraints on K, a, and b so that the system is 
Type 1?
(b) What are the constraints placed on K. a, and b so that the system is both stable and Type 1 ?
(c) What are the constraints on a and b so that the system is both Type 1 and remains stable for
(c) What are the constraints on a and b so that the system is both Type 1 and remains stable for 
every positive value for K?
Step-by-step solution
step 1 of 5
D (s) =
s"+2?s+l 
s+b
Y (s) _________ K(s+a)
E (s ) (s“+2?s+l)(s+b)+K(s+a)
( ! “+2?!+ l)(s+b)
E = R -Y = . , \ ----- -R(s)
(s“+2?s+l)(s+b)+K(s+a)
Step 2 of 5
, . sfs+b)fs^+2^s+ll , . 
a. S.S.E.=limsE(s)= ^ R (0
(s^+2^+l) ( s ^ ) +K (s+a)
Fortype -1 systemJb=0, K ^O , a^O
Step 3 of 5
b. For the given system,
b=0;
R ( s ) = | .
s
and the system shoul d be of type -1. 
S.S.E.=limsEfs)i-»0 ' '
s"(s"+2?s+l) 1
= lim -------- -̂----------- ------ X - r
*-»«(s^+24s+l)s+K(s+a) ŝ
Char, equation = s^+2^^+(K+l)s+Kat*0
Step 4 of 5
For Stability,
1 K+1
s’ Ka
s‘ 0
s" Ka
Ka>0, (K+1) -E i> 0
25
=>Ka0, Ka0, r^>0| a1 ^ |0